Targeting the gradient flow

13 May, 2026

In this post, we will first summarize what we discussed in part 1 and part 2, then add a few extra thoughts.

Summary of the two posts

Part 1: we picked a function $f:ℝ\to ℝ$ and built a simple vector field such that for any trajectory $(x,y):{ℝ}^{>0}\to {ℝ}^2$, the trajectory is attracted to the graph of $f$, in the sense that if $u(t):=y(t)-f(x(t))$, then ${lim}_{t}u(t)=0$. To do this, we defined the residual $u:=y-f(x)$ and forced it to satisfy the ODE $u^{\prime }=-ku,k>0$ (which also gives an ODE for $y$). We also picked an arbitrary direction for $x$ to move with $x^{\prime }=1$. We showed that the corresponding vector field $F(x,y)=(x^{\prime },y^{\prime })$ works for our purposes!

We discussed that we can extend the construction to vector fields on $x\in {ℝ}^n$ as well as the case of $k$ being a function of $x$ instead of constant.

Finally, given an existing vector field $F$ and the assumption $u^{\prime }=-ku$, we showed that $u$ and $F$ satisfy a transport PDE:

$F·\nabla u+ku=0.$

Part 2: we applied some of these ideas to optimization. Starting from a smooth $g:ℝ\to ℝ$ for which we want to find a minimum, we created various vector fields $F$ such that the trajectories following them converge to the stationary points of $g$. We used a similar construction, setting $x^{\prime }=-g^{\prime }(x)$ and getting an ODE for $y$ from $u=y-g(x)$ and $u^{\prime }=-ku$.

We also discovered that we have some freedom in picking $x^{\prime }$, i.e., for any well-behaved $B:{ℝ}^2\to ℝ$, setting $x^{\prime }=-g^{\prime }(x)+B(x,y)u$ will still give us convergent dynamics as long as we are not completely cancelling $-g^{\prime }(x)$ close to the stationary points. This opens up new possibilities, e.g., $B$ can also be a stochastic function, etc.

Follow-up questions and thoughts

It is fun watching the trajectories skip over some of the basins of attraction, but can we actually get more interesting behaviors out of picking the right $B$?

We assumed $u^{\prime }=-ku\Rightarrow u=\exp (-kt)u_0$, and so no trajectory will have non-trivial periodic orbits. If a trajectory did, the residual would also have to be periodic and that is impossible as $\exp (-kt)$ is not a periodic function. The only way for periodicity to hold is if $u(t)=0$ for all $t>t_0$. However, on the attracting manifold we are following GD which also doesn't have any non-trivial periodic orbits.

So, regardless of the choice of $B$ (assuming it is well-behaved), these methods cannot produce sustained exploration. There is an exponential clock, and once it rings, exploration is over: we are back to following gradient descent. We can give more time for exploration by adjusting the decay, e.g., setting $u=u_0/(1+t)$.

We don't have to match $g$.

In part 2, we had the residual be $u=y-g(x)$ and $u^{\prime }=-ku$ (or, equivalently, have $u$ satisfy our transport PDE). However, what if instead of targeting the graph of $g$, we target the gradient flow $-g^{\prime }$ directly?

To do this, we will work on the $(x,v)$ space where $v=x^{\prime }$. We set $u=v-(-g^{\prime }(x))$, i.e., we are getting trajectories to match the gradient flow, $-g^{\prime }(x)$. Now

$u^{\prime }=v^{\prime }+g^{″}(x)x^{\prime }=v^{\prime }+g^{″}(x)v.$

If we now enforce $u^{\prime }=-ku$ (and do a few calculations) we get:

$x^{″}+(k+g^{″}(x))x^{\prime }+k{g}^{\prime }(x)=0.$

So the same residual-decay idea produces second-order optimization dynamics if we focus on gradient flow.

I was wondering if this is a known second-order ODE. It looks like a special case of the form studied by Alvarez¹, et al., : $x^{″}+(\alpha +\beta g^{″}(x))x^{\prime }+g^{\prime }(x)=0$.

There's potential to recover further ODEs from the literature (e.g., with $1/t$ damping) by allowing time-dependence in $k$ and setting $u=v+\eta (t)g^{\prime }(x)$. I leave that for future work.

TL;DR: All this came out of picking a residual form and then attaching a decay assumption.

Footnotes